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Mobility Information Capacity in the Sky: A Gaussian Channel Perspective
Weijie Yuan, Fan Liu, Shuangyang Li, Lin Zhou, Pingzhi Fan
TL;DR
Existing airspace-capacity measures focus on occupancy or flow, although equal vehicle counts can allow different motion alternatives. This paper defines MIC through a lifted linear-Gaussian channel and derives a mode-based capacity with water-filling allocation. Under stated mode-growth and gain conditions, its normalized rate has a Shannon-like form; the measure remains a distinguishability baseline, not a safety-constrained capacity.
Problem
Occupancy- and flow-based metrics do not capture how much motion diversity is available when the same number of vehicles can take different routes or speed profiles.
Source: §I · p. 1Method
The paper models maneuver inputs and trajectory observations as a lifted linear-Gaussian channel, then whitens maneuver cost and observation uncertainty to identify mobility modes and allocate resources.
Source: §I · p. 1, §II · p. 2, §III · p. 3Results
Under linear growth of nondegenerate modes and asymptotically equal gains, normalized capacity has a Shannon-like form; at K=80, full-trajectory and terminal-only rates are approximately 0.132 and 0.027 bits/slot, respectively.
Source: §IV · p. 4, §V · p. 5Takeaways & Limitations
MIC provides a baseline for comparing observable maneuver diversity under specified dynamics, resource budgets, and uncertainty.
Source: §V · p. 5A safety-constrained airspace-capacity measure still requires physical admissibility and multi-vehicle interactions.
Source: §V · p. 5
Abstract
from arXiv · showhide
Existing airspace capacity metrics mainly quantify occupancy or flow, although the same number of aerial vehicles may result in different motion alternatives. This letter establishes \emph{mobility information capacity} as an information-theoretic measure for low-altitude wireless networks. It quantifies the maximum information that trajectory observations reveal about intentional maneuver inputs under a given maneuver-resource budget and environmental uncertainty. For a common fixed feedback architecture, we formulate a lifted linear-Gaussian mobility channel and derive its finite-horizon log-determinant capacity. Cost and uncertainty whitening gives the spatiotemporal mobility eigenmodes, whose optimal maneuver-resource allocation follows water-filling. When the number of nondegenerate modes grows linearly with time and their efficiencies become asymptotically symmetric, we arrive at the Shannon-like law $R_M^{\rm G}=\frac{B_M}{2}\log_2(1+\mathrm{MNR})$, where MNR is the mobility-to-noise ratio. The proposed measure opens a motion-centric capacity perspective for the sky, while remaining a distinguishability baseline rather than a collision- or geometry-constrained airspace capacity.
I. INTRODUCTION
The introduction motivates a motion-centric capacity measure because occupancy and flow do not capture the diversity of feasible routes and speed profiles. It defines MIC through input–trajectory information under a maneuver budget and previews a Gaussian-channel analysis with mode-based allocation and long-horizon results.
I. INTRODUCTION
- Existing airspace metrics emphasize usable volume, vehicle density, or traffic flow, but equal vehicle counts can permit different route choices and speed profiles.This motivates measuring how much freedom of motion remains once vehicles enter a region.Source: §I · p. 1
- The introduction frames mobility capacity as how rapidly distinguishable motion alternatives can grow under maneuver effort and uncertainty in trajectory observations.The uncertainty includes wind, execution error, and sensing noise.Source: §I · p. 1
- MIC is the maximum input–trajectory mutual information under a maneuver-resource budget, complementing occupancy- and flow-based measures rather than determining safe aircraft counts or traffic throughput.The measure is proposed for comparing sensing architectures, maneuver efficiency, and information loss from observed trajectories under a common resource budget.Source: §I · p. 1
- The letter formulates a finite-horizon lifted mobility channel and derives an exact log-determinant MIC for linear-Gaussian dynamics.Whitening maneuver cost and trajectory uncertainty yields mobility modes whose optimal resource allocation follows water-filling.Source: §I · p. 1
II. MOBILITY CHANNEL AND INFORMATION CAPACITY
The section models intentional maneuver sequences as inputs to a finite-horizon Gaussian channel whose outputs are observed trajectories, under a maneuver-resource budget and environmental uncertainty. It defines mobility information capacity as the information trajectory observations reveal about maneuver inputs and states an independence condition for the channel uncertainties.
II. MOBILITY CHANNEL AND INFORMATION CAPACITY
- The finite-horizon Gaussian channel maps exogenous maneuver inputs to trajectory observations, with uncertainty including propagated process, execution, localization, and observation errors.The common feedback controller is fixed across mobility alternatives.Source: §II · p. 2
- Mobility information capacity is the maximum input–trajectory mutual information under a quadratic maneuver-resource budget.The resource rate and horizon determine the total budget; higher weights penalize more costly mobility directions.Source: §II · p. 2
- Under U ⊥ ZK, trajectory information measures variation attributable to intentional maneuvers rather than uncertainty-induced spread.Stronger disturbances can increase trajectory entropy without creating more distinguishable maneuver alternatives.Source: §II · p. 2
- Proposition 1 gives the Gaussian mobility information capacity as a finite-horizon log-determinant expression.The independence assumption applies when maneuvers are selected exogenously; observation-adaptive inputs may require a causal information measure such as directed information.Source: §II · p. 2
Z,K GKKUGT
The capacity derivation reduces optimization to choosing the maneuver-input covariance under the resource constraint. Centering removes the input mean's additional cost, and Gaussian extremality yields the covariance-based log-determinant capacity.
Z,K GKKUGT
- The covariance matrix KU specifies the maneuver-input distribution optimized under the resource constraint.The covariance optimization allocates available maneuver resource across observable mobility directions.Source: Z,K GKKUGT · pp. 2–3
- Centering the maneuver input preserves mutual information while removing its mean-related cost, so an optimal input may be zero mean.The average cost contains both tr(QK KU) and the mean quadratic term.Source: Z,K GKKUGT · p. 2
- Subtracting the known initial-state response leaves mutual information unchanged while retaining motion differences induced by candidate commands.The response is common across maneuver sequences when the initial state and system matrices are known.Source: Z,K GKKUGT · p. 2
- For fixed KU, a zero-mean Gaussian input maximizes output entropy and attains the bound used in the capacity expression.The remaining step is to optimize KU subject to the maneuver-resource constraint.Source: Z,K GKKUGT · p. 3
III. MOBILITY MODE AND SHANNON-LIKE LAW
Cost- and uncertainty-whitening exposes observable maneuver modes, whose resource allocation follows water-filling. With linearly growing useful modes and asymptotically equal gains, the long-horizon information rate has a Shannon-like form.
Mobility modes
- Whitening maneuver cost and trajectory uncertainty maps normalized maneuvers to noise-normalized trajectory variation, revealing the mobility modes.The number of nonzero modes is L(T) = rank(GK).Source: §III · p. 3
- The optimal allocation follows water-filling and can be designed offline from the known finite-horizon model and uncertainty statistics.The eigenmode-coordinate allocation is diagonal at the optimum, and its KKT conditions yield water-filling.Source: §III · p. 3
- Each mode’s gain measures observable trajectory response per unit normalized maneuver resource, while its allocated resource sets that mode’s effective MNR.A zero gain denotes a maneuver pattern with no observable signature in the selected measurements; weak modes may remain inactive.Source: §III · p. 3
Shannon-like law
- When useful modes grow linearly and nonzero gains converge to λ, the long-horizon rate is (BM/2) log2(1 + λPM/BM).BM is the useful-mode creation rate; PM/BM is the asymptotic resource per mode, and λ is the observable response per unit normalized resource.Source: §III · pp. 3–4
- Linear mode growth can arise when each control interval adds maneuver effects retained in the observed trajectory; full-trajectory observation can preserve independent temporal patterns.An example uses double-integrator motion, where each acceleration command contributes new independent directions under full-trajectory observation.Source: §III · pp. 3–4
- A fixed-dimensional terminal observation caps useful modes, whereas periodic sampling can support a positive rate if it retains horizon-proportional modes with gains bounded away from zero.For sliding-window processing, the rate also depends on whether information from earlier windows is preserved.Source: §III · p. 4
IV. NUMERICAL RESULTS
For (14), full-trajectory observations maintain an approximately constant finite-horizon rate, whereas terminal-only rates decay; finite-horizon mode gains also yield selective water-filling allocation.
IV. NUMERICAL RESULTS
- The example models one lateral UAV deviation with xk+1 = 0.85xk + uk + wk and yk = xk + vk.Here wk represents wind or execution uncertainty, and vk represents localization or observation noise.Source: §IV · p. 4
- Earlier maneuver inputs affect more subsequent observations, producing unequal mode gains and zero allocation to seven sufficiently weak modes at K = 20.The example uses the exact water-filling solution for the time-invariant system in (14).Source: Figure 2 · p. 4, §IV · p. 4
- At K = 80, full-trajectory and terminal-only rates are approximately 0.132 and 0.027 bits/slot, respectively.The full-trajectory observer preserves an increasing number of modes, whereas the terminal observer retains only yK.Source: Figure 3 · p. 4, §IV · p. 4
- A positive long-horizon rate requires an observation architecture that preserves a growing number of informative motion modes.The terminal-only rate decays with K, while the full-trajectory rate remains approximately constant in the example.Source: §IV · p. 4
V. EXTENSIONS AND CONCLUSION
The capacity measures distinguishability rather than physical admissibility, and its log-determinant expression may only upper-bound capacity under admissibility constraints. The conclusion gives a Shannon-like rate under specified mode-growth conditions while identifying causal and multi-vehicle settings as extensions.
V. EXTENSIONS AND CONCLUSION
- Under admissibility constraints, Gaussian inputs may be infeasible, so the log-determinant expression generally serves only as an upper bound.The capacity-achieving Gaussian input has unbounded support and may violate actuator, fly-zone boundary, or collision constraints.Source: §V · pp. 4–5
- Causal maneuver inputs may require directed information, while multi-vehicle collision constraints couple feasible maneuver sequences across agents.The formulation assumes a common feedback controller and an exogenously selected maneuver sequence.Source: §V · p. 5
- Under linear mode growth and asymptotically equal gains, normalized capacity takes a Shannon-like form; fixed-dimensional terminal observations have vanishing normalized rate under bounded gains.The observation architecture determines which maneuver modes remain informative as the horizon grows.Source: §V · p. 5
- The measure is a baseline for comparing observable maneuver diversity under specified dynamics, resource budgets, and uncertainty, not a safety-constrained airspace-capacity measure.Physical admissibility and multi-vehicle interactions remain necessary for a safety-constrained measure.Source: §V · pp. 4–5